Insertion of GP's between 2 Numbers

Q. If $m$ is the arithmetic mean of two distinct real numbers $l$ and $n(l,n>1)$ and $G_1,G_2$ and $G_3$ are three geometric means between $l$ and $n,$ then $G_1^4+2G_2^4+G_3^4$ equals

1. $4lm{n}^2$
2. $4l^2{m}^2{n}^2$
3. $4l{m}^{2}n$
4. $4l^{2}mn$

Q.

Insert 6 geometric means between 27 and $\frac{1}{81}$.

Q.

Insert 5 geometric means between 16 and $\frac{1}{4}$.

Q. Let $a_1,a_2,a_3,\dots$ be a G.P. with $a_1=a$ and common ratio $r,$ where $a$ and $r$ positive integers, then the number of ordered pairs $(a,r)$ such that $\sum _{k=1}^{12}{\log }_8{a}_k=2010$ is

Q.

Insert 5 geometric means between $\frac{32}{9}$ and $\frac{81}{2}$.

Q.

Prove that the product of n geometric means between two quantities is equal to the nth power of a geometric means of those two quantities.

Q. The value of $\alpha$ so that the geometric mean of $x$ and $y$, where $x\ne y$ is $\frac{x^{\alpha +2}+y^{\alpha +2}}{x^{\alpha +1}+y^{\alpha +1}}$, is

1. $-\frac{2}{3}$
2. $-\frac{1}{4}$
3. $-\frac{3}{2}$
4. $\frac{7}{6}$

Q. If n geometric means between a and b be $G_1,G_2,.......G_n$ and a geometric mean be G, then the true relation is

1. $G_1.G_2.......G_n=G^n$
2. $G_1.G_2.......G_n=G^{2/n}$
3. $G_1.G_2.......G_n=G$
4. $G_1.G_2.......G_n=G^{1/n}$

Q.

If n geometric means between a and b be $G_1,G_2,....G_n$ and a geometric mean of a and b be G, then the true relation is

1. 

2. 

3. 

4. 

Q. If three geometric means be inserted between 2 and 32, then the third geometric mean will be

1. 8

2. 4

3. 16
4. 12

Q. If m is the A.M. of two distinct real numbers I and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals:

1. $4lm{n}^2$
2. $4l^2{m}^2{n}^2$
3. $4l^{2}mn$
4. $4l{m}^{2}n$

Q. If n geometric means between a and b be $G_1,G_2,.......G_n$ and a geometric mean be G, then the true relation is

1. $G_1.G_2.......G_n=G$
2. $G_1.G_2.......G_n=G^{1/n}$
3. $G_1.G_2.......G_n=G^n$
4. $G_1.G_2.......G_n=G^{2/n}$

Q. Find the sum of 5 geometric means between $\frac{1}{3}$ and 243, by taking common ratio positive.

1. 121
2. 81
3. 126
4. 111

Q. The two geometric means between the number 1 and 64 are
[Kerala (Engg.) 2002]

1. 1 and 64

2. 4 and 16

3. 2 and 16

4. 8 and 16

Q. The value of $\alpha$ so that the geometric mean of $x$ and $y$, where $x\ne y$ is $\frac{x^{\alpha +2}+y^{\alpha +2}}{x^{\alpha +1}+y^{\alpha +1}}$, is

1. $-\frac{2}{3}$
2. $-\frac{1}{4}$
3. $-\frac{3}{2}$
4. $\frac{7}{6}$

Q. If m is the A.M. of two distinct real numbers I and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals:

1. $4lm{n}^2$
2. $4l^2{m}^2{n}^2$
3. $4l^{2}mn$
4. $4l{m}^{2}n$

Q. The two geometric means between the number 1 and 64 are
[Kerala (Engg.) 2002]

1. 4 and 16

2. 2 and 16

3. 8 and 16

4. 1 and 64

Q. Two numbers between 3 and 81 such that the series $3,G_1,G_2,81$ forms a GP are 9 and 27.

1. True
2. False

Q. The common ratio of G.P. $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+.....$ is

1. $\frac{1}{2}$
2. $\frac{1}{3}$
3. $\frac{1}{4}$
4. $\frac{1}{5}$

Q. If m is the AM of two distinct real numbers l and n(l, n > 1) and $G_1,G_2$ and $G_3$ are three geometric means between l and n, then $G_1^4+2G_2^4+G_3^4$ equals

1. $4l^{2}mn$
2. $4l{m}^{2}n$
3. $lm{n}^2$
4. $l^2{m}^2{n}^2$